\(\int \frac {x (1+x) (1+3 x)}{(-1+x)^2 (-1+3 x) \sqrt [3]{-x+x^3}} \, dx\) [14]

Optimal result

Integrand size = 33, antiderivative size = 166 \[ \int \frac {x (1+x) (1+3 x)}{(-1+x)^2 (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\frac {-3 \left (-x+x^3\right )^{2/3}+2 \sqrt {3} \left (1-2 x+x^2\right ) \arctan \left (\frac {\sqrt {3} \left (124616800+1235276981 x+1609127381 x^2\right )[phone] \sqrt {3} (-1+x) \sqrt [3]{-x+x^3}[phone] \sqrt {3} \left (-x+x^3\right )^{2/3}}{-39304000+3108349623 x+2990437623 x^2}\right )-\left (1-2 x+x^2\right ) \log \left (\frac {-1+3 x+3 (-1+x) \sqrt [3]{-x+x^3}-3 \left (-x+x^3\right )^{2/3}}{-1+3 x}\right )}{2 \left (1-2 x+x^2\right )} \] Output:

1/2*(2*3^(1/2)*(x^2-2*x+1)*arctan((612314840*3^(1/2)*(x^3-x)^(1/3)*(-1+x)+ 
3^(1/2)*(1609127381*x^2+1235276981*x[phone])[phone]*3^(1/2)*(x^3-x) 
^(2/3))/(2990437623*x^2+3108349623*x-39304000))-(x^2-2*x+1)*ln((3*(-1+x)*( 
x^3-x)^(1/3)+3*x-3*(x^3-x)^(2/3)-1)/(-1+3*x))-3*(x^3-x)^(2/3))/(x^2-2*x+1)
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int \frac {x (1+x) (1+3 x)}{(-1+x)^2 (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int \frac {x (1+x) (1+3 x)}{(-1+x)^2 (-1+3 x) \sqrt [3]{-x+x^3}} \, dx \] Input:

Integrate[(x*(1 + x)*(1 + 3*x))/((-1 + x)^2*(-1 + 3*x)*(-x + x^3)^(1/3)),x 
]
 

Output:

Integrate[(x*(1 + x)*(1 + 3*x))/((-1 + x)^2*(-1 + 3*x)*(-x + x^3)^(1/3)), 
x]
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x (x+1) (3 x+1)}{(x-1)^2 (3 x-1) \sqrt [3]{x^3-x}} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \int -\frac {x^{2/3} (x+1) (3 x+1)}{(1-3 x) (1-x)^2 \sqrt [3]{x^2-1}}dx}{\sqrt [3]{x^3-x}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \int \frac {x^{2/3} (x+1) (3 x+1)}{(1-3 x) (1-x)^2 \sqrt [3]{x^2-1}}dx}{\sqrt [3]{x^3-x}}\)

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{x^2-1} \int \frac {x^{4/3} (x+1) (3 x+1)}{(1-3 x) (1-x)^2 \sqrt [3]{x^2-1}}d\sqrt [3]{x}}{\sqrt [3]{x^3-x}}\)

\(\Big \downarrow \) 1396

\(\displaystyle -\frac {3 \sqrt [3]{-x-1} \sqrt [3]{1-x} \sqrt [3]{x} \int \frac {x^{4/3} (x+1) (3 x+1)}{(1-3 x) \sqrt [3]{-x-1} (1-x)^{7/3}}d\sqrt [3]{x}}{\sqrt [3]{x^3-x}}\)

\(\Big \downarrow \) 281

\(\displaystyle \frac {3 \sqrt [3]{-x-1} \sqrt [3]{1-x} \sqrt [3]{x} \int \frac {(-x-1)^{2/3} x^{4/3} (3 x+1)}{(1-3 x) (1-x)^{7/3}}d\sqrt [3]{x}}{\sqrt [3]{x^3-x}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 \sqrt [3]{-x-1} \sqrt [3]{1-x} \sqrt [3]{x} \int \left (-\frac {(-x-1)^{2/3} x^{4/3}}{(1-x)^{7/3}}+\frac {2 (-x-1)^{2/3} \sqrt [3]{x}}{3 (1-3 x) (1-x)^{7/3}}-\frac {2 (-x-1)^{2/3} \sqrt [3]{x}}{3 (1-x)^{7/3}}\right )d\sqrt [3]{x}}{\sqrt [3]{x^3-x}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \sqrt [3]{-x-1} \sqrt [3]{1-x} \sqrt [3]{x} \left (\frac {2}{3} \int \frac {(-x-1)^{2/3} \sqrt [3]{x}}{(1-3 x) (1-x)^{7/3}}d\sqrt [3]{x}-\frac {(-x-1)^{2/3} x^{5/3} \operatorname {AppellF1}\left (\frac {5}{3},\frac {7}{3},-\frac {2}{3},\frac {8}{3},x,-x\right )}{5 (x+1)^{2/3}}-\frac {(-x-1)^{2/3} x^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {7}{3},\frac {5}{3},\frac {2 x}{x+1}\right )}{3 (x+1)^{4/3}}\right )}{\sqrt [3]{x^3-x}}\)

Input:

Int[(x*(1 + x)*(1 + 3*x))/((-1 + x)^2*(-1 + 3*x)*(-x + x^3)^(1/3)),x]
 

Output:

$Aborted
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.82 (sec) , antiderivative size = 670, normalized size of antiderivative = 4.04

method	result	size
trager	\(\text {Expression too large to display}\)	\(670\)

Input:

int(x*(1+x)*(1+3*x)/(-1+x)^2/(-1+3*x)/(x^3-x)^(1/3),x,method=_RETURNVERBOS 
E)
 

Output:

-3/2/(-1+x)^2*(x^3-x)^(2/3)-3*ln((4005*RootOf(9*_Z^2-3*_Z+1)^2*x^2+9702*Ro 
otOf(9*_Z^2-3*_Z+1)*(x^3-x)^(2/3)-3321*RootOf(9*_Z^2-3*_Z+1)*(x^3-x)^(1/3) 
*x-10413*RootOf(9*_Z^2-3*_Z+1)^2*x-7716*RootOf(9*_Z^2-3*_Z+1)*x^2-1107*(x^ 
3-x)^(2/3)+3321*RootOf(9*_Z^2-3*_Z+1)*(x^3-x)^(1/3)-2127*(x^3-x)^(1/3)*x+4 
806*RootOf(9*_Z^2-3*_Z+1)^2+13173*RootOf(9*_Z^2-3*_Z+1)*x+3679*x^2+2127*(x 
^3-x)^(1/3)-6963*RootOf(9*_Z^2-3*_Z+1)-2264*x+1981)/(-1+3*x))*RootOf(9*_Z^ 
2-3*_Z+1)-ln(-(-8730*RootOf(9*_Z^2-3*_Z+1)^2*x^2+9702*RootOf(9*_Z^2-3*_Z+1 
)*(x^3-x)^(2/3)-6381*RootOf(9*_Z^2-3*_Z+1)*(x^3-x)^(1/3)*x+22698*RootOf(9* 
_Z^2-3*_Z+1)^2*x-411*RootOf(9*_Z^2-3*_Z+1)*x^2-2127*(x^3-x)^(2/3)+6381*Roo 
tOf(9*_Z^2-3*_Z+1)*(x^3-x)^(1/3)-1107*(x^3-x)^(1/3)*x-10476*RootOf(9*_Z^2- 
3*_Z+1)^2-13947*RootOf(9*_Z^2-3*_Z+1)*x+2264*x^2+1107*(x^3-x)^(1/3)+4512*R 
ootOf(9*_Z^2-3*_Z+1)+1415*x+283)/(-1+3*x))+ln((4005*RootOf(9*_Z^2-3*_Z+1)^ 
2*x^2+9702*RootOf(9*_Z^2-3*_Z+1)*(x^3-x)^(2/3)-3321*RootOf(9*_Z^2-3*_Z+1)* 
(x^3-x)^(1/3)*x-10413*RootOf(9*_Z^2-3*_Z+1)^2*x-7716*RootOf(9*_Z^2-3*_Z+1) 
*x^2-1107*(x^3-x)^(2/3)+3321*RootOf(9*_Z^2-3*_Z+1)*(x^3-x)^(1/3)-2127*(x^3 
-x)^(1/3)*x+4806*RootOf(9*_Z^2-3*_Z+1)^2+13173*RootOf(9*_Z^2-3*_Z+1)*x+367 
9*x^2+2127*(x^3-x)^(1/3)-6963*RootOf(9*_Z^2-3*_Z+1)-2264*x+1981)/(-1+3*x))
 

Fricas [A] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.88 \[ \int \frac {x (1+x) (1+3 x)}{(-1+x)^2 (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\frac {2 \, \sqrt {3} {\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {612314840 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (1609127381 \, x^{2} + 1235276981 \, x + 124616800\right )} + 2605939922 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{2990437623 \, x^{2} + 3108349623 \, x - 39304000}\right ) - {\left (x^{2} - 2 \, x + 1\right )} \log \left (\frac {3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 3 \, x - 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} - 1}{3 \, x - 1}\right ) - 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} \] Input:

integrate(x*(1+x)*(1+3*x)/(-1+x)^2/(-1+3*x)/(x^3-x)^(1/3),x, algorithm="fr 
icas")
 

Output:

1/2*(2*sqrt(3)*(x^2 - 2*x + 1)*arctan((612314840*sqrt(3)*(x^3 - x)^(1/3)*( 
x - 1) + sqrt(3)*(1609127381*x^2 + 1235276981*x + 124616800) + 2605939922* 
sqrt(3)*(x^3 - x)^(2/3))/(2990437623*x^2 + 3108349623*x - 39304000)) - (x^ 
2 - 2*x + 1)*log((3*(x^3 - x)^(1/3)*(x - 1) + 3*x - 3*(x^3 - x)^(2/3) - 1) 
/(3*x - 1)) - 3*(x^3 - x)^(2/3))/(x^2 - 2*x + 1)
 

Sympy [F]

\[ \int \frac {x (1+x) (1+3 x)}{(-1+x)^2 (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int \frac {x \left (x + 1\right ) \left (3 x + 1\right )}{\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right )^{2} \cdot \left (3 x - 1\right )}\, dx \] Input:

integrate(x*(1+x)*(1+3*x)/(-1+x)**2/(-1+3*x)/(x**3-x)**(1/3),x)
 

Output:

Integral(x*(x + 1)*(3*x + 1)/((x*(x - 1)*(x + 1))**(1/3)*(x - 1)**2*(3*x - 
 1)), x)
 

Maxima [F]

\[ \int \frac {x (1+x) (1+3 x)}{(-1+x)^2 (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int { \frac {{\left (3 \, x + 1\right )} {\left (x + 1\right )} x}{{\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (3 \, x - 1\right )} {\left (x - 1\right )}^{2}} \,d x } \] Input:

integrate(x*(1+x)*(1+3*x)/(-1+x)^2/(-1+3*x)/(x^3-x)^(1/3),x, algorithm="ma 
xima")
 

Output:

integrate((3*x + 1)*(x + 1)*x/((x^3 - x)^(1/3)*(3*x - 1)*(x - 1)^2), x)
 

Giac [F]

\[ \int \frac {x (1+x) (1+3 x)}{(-1+x)^2 (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int { \frac {{\left (3 \, x + 1\right )} {\left (x + 1\right )} x}{{\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (3 \, x - 1\right )} {\left (x - 1\right )}^{2}} \,d x } \] Input:

integrate(x*(1+x)*(1+3*x)/(-1+x)^2/(-1+3*x)/(x^3-x)^(1/3),x, algorithm="gi 
ac")
 

Output:

integrate((3*x + 1)*(x + 1)*x/((x^3 - x)^(1/3)*(3*x - 1)*(x - 1)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x (1+x) (1+3 x)}{(-1+x)^2 (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=\int \frac {x\,\left (3\,x+1\right )\,\left (x+1\right )}{{\left (x^3-x\right )}^{1/3}\,\left (3\,x-1\right )\,{\left (x-1\right )}^2} \,d x \] Input:

int((x*(3*x + 1)*(x + 1))/((x^3 - x)^(1/3)*(3*x - 1)*(x - 1)^2),x)
 

Output:

int((x*(3*x + 1)*(x + 1))/((x^3 - x)^(1/3)*(3*x - 1)*(x - 1)^2), x)
 

Reduce [F]

\[ \int \frac {x (1+x) (1+3 x)}{(-1+x)^2 (-1+3 x) \sqrt [3]{-x+x^3}} \, dx=3 \left (\int \frac {x^{3}}{3 x^{\frac {10}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}-7 x^{\frac {7}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}+5 x^{\frac {4}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}-x^{\frac {1}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}}d x \right )+4 \left (\int \frac {x^{2}}{3 x^{\frac {10}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}-7 x^{\frac {7}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}+5 x^{\frac {4}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}-x^{\frac {1}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}}d x \right )+\int \frac {x}{3 x^{\frac {10}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}-7 x^{\frac {7}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}+5 x^{\frac {4}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}-x^{\frac {1}{3}} \left (x^{2}-1\right )^{\frac {1}{3}}}d x \] Input:

int(x*(1+x)*(1+3*x)/(-1+x)^2/(-1+3*x)/(x^3-x)^(1/3),x)
 

Output:

3*int(x**3/(3*x**(1/3)*(x**2 - 1)**(1/3)*x**3 - 7*x**(1/3)*(x**2 - 1)**(1/ 
3)*x**2 + 5*x**(1/3)*(x**2 - 1)**(1/3)*x - x**(1/3)*(x**2 - 1)**(1/3)),x) 
+ 4*int(x**2/(3*x**(1/3)*(x**2 - 1)**(1/3)*x**3 - 7*x**(1/3)*(x**2 - 1)**( 
1/3)*x**2 + 5*x**(1/3)*(x**2 - 1)**(1/3)*x - x**(1/3)*(x**2 - 1)**(1/3)),x 
) + int(x/(3*x**(1/3)*(x**2 - 1)**(1/3)*x**3 - 7*x**(1/3)*(x**2 - 1)**(1/3 
)*x**2 + 5*x**(1/3)*(x**2 - 1)**(1/3)*x - x**(1/3)*(x**2 - 1)**(1/3)),x)